Homogenization for non-self-adjoint periodic elliptic operators on an infinite cylinder

الأصل · EN

We consider the problem of homogenization for non-self-adjoint second-order elliptic differential operators Aε of divergence form on L₂(Rᵈ₁ᵈ₂), where d₁ is positive and d₂ is non-negative. The coefficients of the operator Aε are periodic in the first variable with period ε and smooth in a certain sense in the second. We show that, as ε gets small, (Aε-μ)⁻¹ and Dₓ₂(Aε-μ)⁻¹ converge in the operator norm to, respectively, (A⁰-μ)⁻¹ and Dₓ₂(A⁰-μ)⁻¹, where A⁰ is an operator whose coefficients depend only on x₂. We also obtain an approximation for Dₓ₁(Aε-μ)⁻¹ and find the next term in the approximation for (Aε-μ)⁻¹. Estimates for the rates of convergence and the rates of approximation are provided and are sharp with respect to the order.